The compact picture of symmetry breaking operators for rank one - PowerPoint PPT Presentation

The compact picture of symmetry breaking operators for rank one orthogonal and unitary groups Jan Möllers (FAU Erlangen-Nürnberg) joint w/ Bent Ørsted 50th Seminar Sophus Lie, Bedlewo September 26, 2016 Jan Möllers Symmetry breaking operators September 26, 2016 1 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . In the category of smooth admissible Fréchet representations of moderate growth: Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . In the category of smooth admissible Fréchet representations of moderate growth: 1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim Hom G ′ ( π | G ′ , τ ) < ∞ for all irreducible π , τ if the homogeneous space ( G × G ′ ) / diag ( G ′ ) is real spherical. Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . In the category of smooth admissible Fréchet representations of moderate growth: 1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim Hom G ′ ( π | G ′ , τ ) < ∞ for all irreducible π , τ if the homogeneous space ( G × G ′ ) / diag ( G ′ ) is real spherical. 2 (Sun–Zhu 2012) dim Hom G ′ ( π | G ′ , τ ) ≤ 1 for all irreducible π , τ if ( G , G ′ ) is one of the multiplicity-one pairs ( GL ( n , R ) , GL ( n − 1 , R )) , ( O ( p , q ) , O ( p , q − 1 )) , ( U ( p , q ) , U ( p , q − 1 )) , . . . Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . In the category of smooth admissible Fréchet representations of moderate growth: 1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim Hom G ′ ( π | G ′ , τ ) < ∞ for all irreducible π , τ if the homogeneous space ( G × G ′ ) / diag ( G ′ ) is real spherical. 2 (Sun–Zhu 2012) dim Hom G ′ ( π | G ′ , τ ) ≤ 1 for all irreducible π , τ if ( G , G ′ ) is one of the multiplicity-one pairs ( GL ( n , R ) , GL ( n − 1 , R )) , ( O ( p , q ) , O ( p , q − 1 )) , ( U ( p , q ) , U ( p , q − 1 )) , . . . Question 2 , for which π, τ is the multiplicity = 1? In case If the multiplicity is = 1, construct explicitly a symmetry breaking operator π | G ′ → τ . Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Symmetry breaking operators Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup. Definition (Symmetry breaking operator) A symmetry breaking operator is a (continuous) intertwining operator from a representation π of G to a representation τ of G ′ , intertwining for the subgroup G ′ : Hom G ′ ( π | G ′ , τ ) . In the category of smooth admissible Fréchet representations of moderate growth: 1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim Hom G ′ ( π | G ′ , τ ) < ∞ for all irreducible π , τ if the homogeneous space ( G × G ′ ) / diag ( G ′ ) is real spherical. 2 (Sun–Zhu 2012) dim Hom G ′ ( π | G ′ , τ ) ≤ 1 for all irreducible π , τ if ( G , G ′ ) is one of the multiplicity-one pairs ( GL ( n , R ) , GL ( n − 1 , R )) , ( O ( p , q ) , O ( p , q − 1 )) , ( U ( p , q ) , U ( p , q − 1 )) , . . . Question 2 , for which π, τ is the multiplicity = 1? In case If the multiplicity is = 1, construct explicitly a symmetry breaking operator π | G ′ → τ . Idea: Study this question algebraically � in a different category. Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10

Harish-Chandra modules Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup. Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10

Harish-Chandra modules Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup. Construction (Harish-Chandra module) To each smooth admissible representation ( π, V ) of G one can associate the Harish-Chandra module ( π HC , V HC ) of ( g , K ) on � V HC = { v ∈ V : dim span π ( K ) v < ∞} ≃ [ π | K : σ ] · σ ( K -finite vectors in V ). � �� � σ ∈ � K < ∞ Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10

Harish-Chandra modules Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup. Construction (Harish-Chandra module) To each smooth admissible representation ( π, V ) of G one can associate the Harish-Chandra module ( π HC , V HC ) of ( g , K ) on � V HC = { v ∈ V : dim span π ( K ) v < ∞} ≃ [ π | K : σ ] · σ ( K -finite vectors in V ). � �� � σ ∈ � K < ∞ Theorem (Casselman–Wallach) The functor π �→ π HC is an equivalence of categories: � � smooth admissible Fréchet representations → { Harish-Chandra modules } of moderate growth Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10

Harish-Chandra modules Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup. Construction (Harish-Chandra module) To each smooth admissible representation ( π, V ) of G one can associate the Harish-Chandra module ( π HC , V HC ) of ( g , K ) on � V HC = { v ∈ V : dim span π ( K ) v < ∞} ≃ [ π | K : σ ] · σ ( K -finite vectors in V ). � �� � σ ∈ � K < ∞ Theorem (Casselman–Wallach) The functor π �→ π HC is an equivalence of categories: � � smooth admissible Fréchet representations → { Harish-Chandra modules } of moderate growth Problem For a smooth admissible representation π of G the restriction π | G ′ is in general not admissible. � injective map: Hom G ′ ( π | G ′ , τ ) ֒ → Hom ( g ′ , K ′ ) ( π HC | ( g ′ , K ′ ) , τ HC ) . Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10

Harish-Chandra modules Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup. Construction (Harish-Chandra module) To each smooth admissible representation ( π, V ) of G one can associate the Harish-Chandra module ( π HC , V HC ) of ( g , K ) on � V HC = { v ∈ V : dim span π ( K ) v < ∞} ≃ [ π | K : σ ] · σ ( K -finite vectors in V ). � �� � σ ∈ � K < ∞ Theorem (Casselman–Wallach) The functor π �→ π HC is an equivalence of categories: � � smooth admissible Fréchet representations → { Harish-Chandra modules } of moderate growth Problem For a smooth admissible representation π of G the restriction π | G ′ is in general not admissible. � injective map: Hom G ′ ( π | G ′ , τ ) ֒ → Hom ( g ′ , K ′ ) ( π HC | ( g ′ , K ′ ) , τ HC ) . Remark: Surjectivity of this map is equivalent to the automatic continuity of invariant distribution vectors for the homogeneous space ( G × G ′ ) / diag ( G ′ ) . Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10

Degenerate principal series representations Question Construct and classify algebraic symmetry breaking operators in Hom ( g ′ , K ′ ) ( π HC | ( g ′ , K ′ ) , τ HC ) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs ( G , G ′ ) . Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10

Degenerate principal series representations Question Construct and classify algebraic symmetry breaking operators in Hom ( g ′ , K ′ ) ( π HC | ( g ′ , K ′ ) , τ HC ) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs ( G , G ′ ) . Casselman Embedding Theorem Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10